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\begin{document}

\title{Degradation Modeling}
\author[Steven Rinaldo]{Steven G. Rinaldo\inst{1,2}\and Wendy Lee\inst{2}\and J\"{u}rgen Stumper\inst{2}\and
Michael Eikerling\inst{1}}
\institute[shortinst]{\inst{1} Simon Fraser University, 8888 University Drive, Burnaby, B.C. Canada V5A 1S6 \and %
                      \inst{2} Automotive Fuel Cell Cooperation Corporation, 9000 Glenlyon Parkway Burnaby, B.C. Canada V5J 5J8}

\frame{
	\titlepage
}

\section[Outline]{}
\frame{\tableofcontents}


\section{Diagnostic Utilities of the GDM}
\subsection{normalized mean radius and surface area}
\frame{\frametitle{normalized mean radius and surface area}
\begin{figure}
  % Requires \usepackage{graphicx}
  \includegraphics[scale=0.8]{figure4a}\\
  \caption{Relationship between $\bar{r}_N$ and $S_N$ for various $\alpha$}
  \label{fig1}
\end{figure}


\begin{itemize}
\item $\alpha=K_{RDP}m_{0}^{Pt}M_{Pt}/K_{DIS}C_{Pt}^{\infty}L$
\end{itemize}
}

\frame{\frametitle{normalized mean radius and surface area}
\begin{figure}
  % Requires \usepackage{graphicx}
  \includegraphics[scale=0.8]{figure4b}\\
  \caption{Relationship between $\bar{r}_N$ and $S_N$ for various $\alpha$}
  \label{fig1}
\end{figure}


\begin{itemize}
\item experimental correlations are consistent
\end{itemize}
}

\subsection{scaling of $S_N^{\min}$ and $\alpha$}
\frame{\frametitle{scaling of $S_N^{\min}$ and $\alpha$}
\begin{figure}
  % Requires \usepackage{graphicx}
  \includegraphics[scale=0.8]{figure5a}\\
  \caption{Theoretical relationship between $\alpha$ and $S_N^{\min}$}
  \label{fig1}
\end{figure}

\begin{itemize}
\item slope of $\approx0.54$ for $S_N^{\min}\Rightarrow\alpha\propto m_0^{Pt}$
\end{itemize}
}

\frame{\frametitle{scaling of $S_N^{\min}$ and $\alpha$}
\begin{figure}
  % Requires \usepackage{graphicx}
  \includegraphics[scale=0.8]{figure5b}\\
  \caption{Experimental relationship between $m_0^{Pt}$ and $S_N^{\min}$}
  \label{fig1}
\end{figure}


\begin{itemize}
\item slope of $\approx0.58\Rightarrow$ evidence of scaling relationship?
\end{itemize}
}

\subsection{summary}
\frame{\frametitle{systematic experimental studies}
\begin{itemize}
\item variation of initial PRD
\item variation of catalyst composition (ionomer, thickness)
\item effect of loading 
\item how do these effect $\alpha$?
\end{itemize}
}

\frame{\frametitle{model development}
\begin{itemize}
\item study coagulation, detachment
\item Pt effluence
\item spatially varying reaction conditions
\item influence on moment-comparison plots?
\end{itemize}
}


\section{Overview of Oxidation Modeling}
\subsection{modeling approaches}
\frame{\frametitle{modeling approaches}
\begin{block}<1-8>{surface based models}
\begin{itemize}
\item <2-8> oxide layer is infinitely thin
\item <3-8> interfaces: (metal$|$solution)
\item <4-8> electric field effects described phenomenologically

\end{itemize}
\end{block}

\begin{block}<5-8>{point defect models}
\begin{itemize}
\item <6-8> oxide layer considered explicitly
\item <7-8> interfaces: (metal$|$solution), (metal$|$oxide), (oxide$|$solution)
\item <8> electric field effects described self-consistently
\end{itemize}
\end{block}
}

\subsection{modeling approaches evaluation}
\frame{\frametitle{modeling approaches evaluation}
\begin{block}<1-7>{surface based models (advantages and disadvantages)}
\begin{itemize}
\item <2-7> simple formulation; easy to implement (advantage)
\item <3-7> empirical in nature - limited applicability (disadvantage)


\end{itemize}
\end{block}

\begin{block}<4-7>{point defect models (advantages and disadvantages)}
\begin{itemize}
\item <5-7> more involved implementation (disadvantage)
\item <6-7> self-consistent - larger scope (advantage)
\end{itemize}
\end{block}

\begin{itemize}
\item <7> short-term (surface) and long-term (point-defect) approaches
\item <7> surface approaches currently implemented at AFCC
\item <7> refine with long-term approaches as they develop
\end{itemize}
}

\section{Surface Based Model}
\subsection{Pourbaix's equations}
\frame{\frametitle{Pourbaix's equations}
\begin{eqnarray}
% \nonumber to remove numbering (before each equation)
   &Pt+H_2O=Pt(OH)_2+H^++2e^-&  \\
   &Pt(OH_2)+H_2O=PtO_2+2H^++2e^-&  \\
   &PtO_2+H_2O=PtO_3+2H^{+}+2e^-&  \\
   &PtO+2H^+=Pt^{2+}+H_2O&  \\
   &Pt=Pt^{2+}+2e^-&  \\
   &Pt^{2+}+2H_2O=PtO_2+4H^{+}+2e^-&
\end{eqnarray}

\begin{itemize}
\item empirical thermodynamic relationships
\item kinetic analogs of the thermodynamic relationships?
\item is this a good description of oxidation processes for Pt?
\end{itemize}
}

\subsection{kinetic surface model}
\frame{\frametitle{kinetic surface model}
\begin{itemize}
\item fractional coverages: $\theta_{Pt}(t), \theta_{Pt(OH)_2}(t), \theta_{PtO_2}(t), \theta_{PtO_3}(t)$
\item Pt concentration: $C_{Pt^{2+}}(t)$
\item five differential rate equations
\item assume redeposition is negligible (low Pt concentration)
\item determine response of above functions to $\Phi_M(t)$
\end{itemize}
}

\subsection{kinetic surface model: test output}
\frame{\frametitle{kinetic surface model: test output}
\begin{figure}
  % Requires \usepackage{graphicx}
  \includegraphics[scale=0.8]{figure1}\\
  \caption{triangle wave potential perturbation ($\Phi_M(t)$).}\label{fig1}
\end{figure}
}


\frame{\frametitle{kinetic surface model: test output}
\begin{figure}
  % Requires \usepackage{graphicx}
  \includegraphics[scale=0.8]{figure1a}\\
  \caption{surface coverages: triangle wave potential perturbation.}\label{fig1}
\end{figure}
}

\frame{\frametitle{kinetic surface model: test output}
\begin{figure}
  % Requires \usepackage{graphicx}
  \includegraphics[scale=0.8]{figure1b}\\
  \caption{current density: triangle wave potential perturbation.}\label{fig1}
\end{figure}
}



\subsection{kinetic surface model: initial implementation}
\frame{\frametitle{kinetic surface model: initial implementation}
% Define block styles

\tikzstyle{block} = [draw, fill=black!20, rectangle,
    minimum height=3em, minimum width=6em]
\tikzstyle{sum} = [draw, fill=black!20, circle, node distance=1cm]
\tikzstyle{input} = [coordinate]
\tikzstyle{output} = [coordinate]
\tikzstyle{pinstyle} = [pin edge={to-,thin,black}]

\begin{center}
% The block diagram code is probably more verbose than necessary
\begin{tikzpicture}[auto, node distance=2cm,>=latex']
    % We start by placing the blocks
    \node [input, name=input] {};
    \node [sum, right of=input] (sum) {};
    \node [block, right of=sum] (controller) {$\Phi_M(t)$};
    \node [block, right of=controller, node distance=3cm] (system) {oxidation};
    \node [block, above of=system] (calibration) {calibration};
    % We draw an edge between the controller and system block to
    % calculate the coordinate u. We need it to place the measurement block.
    \draw [->] (controller) -- node[name=u] {} (system);
    \node [output, right of=system] (output) {};
    \node [block, below of=u] (measurements) {degradation};

    % Once the nodes are placed, connecting them is easy.
    \draw [->] (calibration) -- node {$a_i$} (system);
    \draw [draw,->] (input) -- node {$j(t)$} (sum);
    \draw [->] (sum) -- node {} (controller);
    \draw [-] (system) -- node [name=y] {}(output);
    \draw [->] (output) |- (measurements);
    \draw [->] (measurements) -| node[pos=0.99] {}
        node [near end] {} (sum);
\end{tikzpicture}
\end{center}
}

\subsection{kinetic surface model: calibration}
\frame{\frametitle{kinetic surface model: calibration}
\begin{figure}
  % Requires \usepackage{graphicx}
  \includegraphics[scale=0.8]{figure2}\\
  \caption{Pt concentration: triangle wave potential perturbation.}\label{fig1}
\end{figure}
\begin{itemize}
\item relevance of cathodic dissolution?
\end{itemize}
}


\frame{\frametitle{kinetic surface model: calibration}
\begin{figure}
  % Requires \usepackage{graphicx}
  \includegraphics[scale=0.8]{figure2a}\\
  \caption{Pt concentration: triangle wave potential perturbation.}\label{fig1}
\end{figure}

\begin{itemize}
\item tune rate constants to match experimental kinetics
\end{itemize}
}

\subsection{kinetic surface model: summary}
\frame{\frametitle{kinetic surface model: summary}
\begin{itemize}
\item need more refined approach to oxidation (CV correlation)
\item relate $\Phi_M$ to electric field and oxidation kinetics
\item systematic experimental studies (Pt dissolution, EQCN)
\item rudimentary implementation as drive-cycle degradation test
\item relation to transient performance
\end{itemize}
}

\end{document}
